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Ariftipyus ^hiic^hus hccratkus, naafra^io cum efectiis ad^Ahcrdienjuim 
Utus ofumadvert^t Gewnttrica fcmmaia cUfcnpta, excUmain^e ad 
cainitcs ita dicitur, Bene fperemu^, Hominum enim veili^a video. 

^ttruv. JrcliUcff Uk b. 9nrf. 



w 



PIONEERS OF PROGRESS 

MEN OF SCIENCE 

Edited by S. CHAPMAN, M.A., D.Sc, F.R.S. 



ARCHIMEDES 



BY \J^V 

Sir THOMAS^^ heath 

K.C.B., K.C.V.O., F.R.S. ; Sc.D., Camb. 
Hon. D.Sc, Oxford 



A6s fioi wov ffrS, Ka\ Ktpm r^iv y^v * < | A Q ll 



LONDON: 

SOCIETY FOR PROMOTING 

CHRISTIAN KNOWLEDGE 

NEW YORK: THE MACMILLAN CO. 

1920 



CONTENTS. 

C"*'- PAGE 

I. Archimedes ' j 

II. Greek Geometry to Archimedes n 

III. The Works of Archimedes 24 

IV. Geometry in Archimedes ....... 29 

V. The Sandreckoner ^5 

VI. Mechanics 50 

VII. Hydrostatics 53 

Bibliography <7 

Chronology eg 



CHAPTER I. 

ARCHIMEDES. 

iT) \ If the ordinary person were asked to say off-hand what 
he knew of Archimedes, he would probably, at the most, 
be able to quote one or other of the well-known stories 
about him : how, after discovering the solution of some 
problem in the bath, he was so overjoyed that he ran 
naked to his house, shouting evpr^Ka, evprjKa (or, as we 
might say, " I've got it, I've got it ") ; or how he said 
** Give me a place to stand on and I will move the 
earth"; or again how he was killed, at the capture of 
Syracuse in the Second Punic War, by a Roman soldier 
who resented being told to get away from a diagram 
drawn on the ground which he was studying. \ 

And it is to be feared that few who are not experts in 
the history of mathematics have any acquaintance with 
the details of the original discoveries in mathematics 
of the greatest mathematician of antiquity, perhaps the 
greatest mathematical genius that the world has ever 
seen. 

History and tradition know Archimedes almost ex- 
clusively as the inventor of a number of ingenious 
mechanical appliances, things which naturally appeal 
more to the popular imagination than the subtleties of 
pure mathematics. • 

Almost all that is told of Archimedes reaches us 
through the accounts by Polybius and Plutarch of the 
siege of Syracuse by Marcellus. He perished in the sack 
of that city in 212 B.C., and, as he was then an old man 

(I) 



2 ARCHIMEDES 

(perhaps 7 5 yea rs old), he must have been born about 
287 B.C. [He was the son of Phidias, an astronomer, 
and was a friend and kinsman of King Hieron of Syracuse 
and his son Gelon. He spent some time at Alexandria 
studying with the successors of Euclid (Euclid who 
flourished about 300 B.C. was then no longer living). 
It was doubtless at Alexandria that he made the 
acquaintance of Conon of Samos, whom he admired 
as a mathematician and cherished as a friend, as well 
as of Eratosthenes ; .to the former, and to the latter 
during his early perio^he was in the habit of communi- 
cating his discoveries before their publication. It was 
also probably in Egypt that he invented the water-screw 
known by his name, the immediate purpose being the 
drawing of water for irrigating fields. 
r After his return to Syracuse he lived a life entirely 
devoted to mathematical research. Incidentally he be- 
came famous through his clever mechanical inventions. 
"^CThese things were, however, in his case the " diversions 
of geometry at play," and he attached no importance to 
them. In the words of Plutarch, " he possessed so lofty 
a spirit, so profound a soul, and such a wealth of scientific 
knowledge that, although these inventions had won for 
him the renown of more than human sagacity, yet he 
would not consent to leave behind him any written work 
on such subjects, but, regarding as ignoble and sordid 
the business of mechanics and every sort of art which is 
directed to practical utility, he placed his whole ambition 
in those speculations in the beauty and subtlety of which 
there is no admixture of the common needs of life ". 
• During the siege of Syracuse Archimedes contrived 
all sorts of engines against the Roman besiegers. There 
were catapults so ingeniously constructed as to be 
equally serviceable at long or short range, and machines 
for discharging showers of missiles through holes made 
in the walls. Other machines consisted of long mov- 
able poles projecting beyond the walls ; some of these 



Archimedes . 3 

dropped heavy weights upon the enemy's ships and on 
the constructions which they called sambuca, from their 
resemblance to a musical instrument of that name, and 
which consisted of a protected ladder with one end 
resting on two quinqueremes lashed together side by 
side as base, and capable of being raised by a windlass ; 
others were fitted with an iron hand or a beak like that 
of a crane, which grappled the prows of ships, then 
lifted them into the air and let them fall again, j Marcellus 
is said to have derided his own engineers and artificers with 
the words, " Shall we not make an end of fighting with 
this geometrical Briareus who uses our ships like cups to 
ladle water from the sea, drives our sambuca off ignomini- 
ously with cudgel-blows, and, by the multitude of missiles 
that he hurls at us all at once, outdoes the hundred- 
handed giants of mythology ? " But the exhortation had 
no effect, the Romans being in such abject terror that, 
"if they did but see a piece of rope or wood projecting 
above the wall they would cry * there it is,' declaring 
that Archimedes was setting some engine in motion 
against them, and would turn their backs and run away, 
insomuch that Marcellus desisted from all fighting and 
a.ssault, putting all his hope in a long siege". 
^i Archimedes died, as he had lived, absorbed in mathe- 
matical contemplation. The accounts of the circum- 
stances of his death differ in some details. Plutarch 
gives more than one version in the following passage : 
'* Marcellus was most of all afflicted at the death of 
Archimedes, for, as fate would have it, he was intent on 
working out some problem with a diagram, and, his mind 
and his eyes being alike fixed on his investigation, jjie 
never noticed the incursion of the Romans nor the 
capture of the city. And when a soldier came up to 
him suddenly and bade him follow to Marcellus, he 
refused to do so until he had worked out his problem to 
a demonstration ; whereat the soldier was so enraged 
that he drew his sword and slew him.'\ Others say that 

i ' ■ ..*4 



4 ARCHIMEDES 

the Roman ran up to him with a drawn sword, threaten- 
ing to kill him ; and, when Archimedes saw him, he 
begged him earnestly to wait a little while in order that 
he might not leave his problem incomplete and unsolved, 
but the other took no notice and killed him. Again, 
there is a third account to the effect that, as he was 
carrying to Marcellus some of his mathematical instru- 
ments, sundials, spheres, and angles adjusted to the 
apparent size of the sun to the sight, some soldiers met 
him and, being under the impression that he carried gold 
in the vessel, killed him." The most picturesque version 
of the story is that which represents him as saying to a 
Roman soldier who came too close, '* Stand away, fellow, 
from my diagram," whereat the man was so enraged that 
he killed him. 

ypVrchimedes is said to have requested his friends and 
" relatives to place upon his tomb a representation of a 
cylinder circumscribing a sphere within it, together with 
an inscription giving the ratio (3/2) which the cylinder 
bears to the sphere ; from which we may infer that he 
himself regarded the discovery of this ratio as his greatest 
achievement.^ Cicero, when quaestor in Sicily, found the 
tomb in a neglected state and restored it. In modern 
times not the slightest trace of it has been found. 

*Beyond the above particulars of the life of Archimedes, 
we have nothing but a number of stories which, if perhaps 
not literally accurate, yet help us to a conception of the 
personality of the man which we would not willingly 
have altered. Thus, in illustration of his entire preoccu- 
pation by his abstract studies, we are told that he would 
forget all about his food and such necessities of life, and 
would be drawing geometrical figures in the ashes of the 
fire, or, when anointing himself, in the oil on his body. 
XDf the same kind is the story mentioned above, that, 
having discovered while in a bath the solution of the 
question referred to him by Hieron as to whether a cer- 
tain crown supposed to have been made of gold did not 



ARCHIMEDES S 

in fact contain a certain proportion of silver, he ran naked 
through the street to his home shouting evprjKa, evprjKa. | ' 
fflt was in connexion with his discovery of the solution ^ 
of the problem To move a given weight by a given force 
that Archimedes uttered the famous saying, ''Give me a 
place to stand on, and I can move the earth " (5o9 //-ot 
TToO arSi Koi kcvcj t^i/ 7^1/, or in his broad Doric, as one 
version has it, ird /3w Kal kcvco rav yav). Plutarch repre- 
sents him as declaring to Hieron that any given weight 
could be moved by a given force, and boasting, in re- 
liance on the cogency of his demonstration, that, if he 
were given another earth, he would cross over to it and 
move this one. "And when Hieron was struck with 
amazement and asked him to reduce the problem to 
practice and to show him some great weight moved by_a Ix* 
small force, he fixed on a ship of burden with three masts 
from the king's arsenal which had only been drawn up 
by the great labour of many men ; and loading her with 
many passengers and a full freight, sitting himself the 
while afar off, with no great effort but quietly setting in 
motion with his hand a compound pulley, he drew the 
ship towards him smoothly and safely as if she were 
moving through the sea." Hieron, we are told elsewhere, 
was so much astonished that he declared that, from that 
day forth, Archimedes's word was to be accepted on 
every subject I „ Another version of the story describes 
the machine used as a helix ; this term must be supposed 
to refer to a screw in the shape of a cylindrical helix 
turned by a handle and acting on a cog-wheel with 
oblique teeth fitting on the screw. 

•Another invention was that of a sphere constructed so 
as to imitate the motions of the sun, the moon, and the 
five planets in the heavens. Cicero actually saw this 
contrivance, and he gives a description of it, stating that 
it represented the periods of the moon and the apparent 
motion of the sun with such accuracy that it would even 
(over a short period) show the eclipses of the sun and 



6 ARCHIMEDES 

moonj It may have been moved by water, for Pappus 
speaks in one place of" those who understand the making 
of spheres and produce a model of the heavens by means 
of the regular circular motion of water '*. In any case it is 
certain that Archimedes was much occupied with astron- 
omy. Livy calls him "unicus spectator caeli siderum- 
que ". Hipparchus says, " From these observations it 
is clear that the differences in the years are altogether 
small, but, as to the solstices, I almost think that both 
I and Archimedes have erred to the extent of a quarter 
of a day both in observation and in the deduction there- 
from".'^ It appears, therefore, that Archimedes had con- 
sidered the question of the length of the year. Macrobius 
says that he discovered the distances of the planets. 
Archimedes himself describes in the Sandreckoner the 
apparatus by which he measured the apparent diameter 
of the sun, i.e. the angle subtended by it at the eye. 
TCXhe story that he set the Roman ships on fire by an 
arrangement of burning-glasses or concave mirrors is not 
found in any authority earlier than Lucian (second cen- 
tury A.D.); but there is no improbability in the idea 
that he discovered some form of burning-mirror, e.g. 
a paraboloid of revolution, which would reflect to one 
point all rays falling on its concave surface in a direction 
parallel to its axis. 



CHAPTER 11. 
GREEK GEOMETRY TO ARCHIMEDES. 

r 

In order to enable the reader to arrive at a correct 
understanding of the place of Archimedes and of the 
significance of his work it is necessary to pass in review 
the course of development of Greek geometry from its 
first beginnings down to the time of Euclid and 
Archimedes. \ 

Greek authors from Herodotus downwards agree in 
saying that geometry was invented by the Egyptians 
and that it came into Greece from Egypt. One account 
says : — 

T" Geometry is said by many to have been invented 
among the Egyptians^ its origin being due to the 
measurement of plots oTland. This was necessary there 
because of the rising of the Nile, which obliterated the 
boundaries appertaining to separate owners. Nor is it 
marvellous that the discovery of this and the other 
sciences should have arisen from such an occasion, since 
everything which moves in the sense of development 
will advance from the imperfect to the perfect. From 
sense-perception to reasoning, and from reasoning to 
understanding, is a natural transition. Just as among 
the Phoenicians, through commerce and exchange, an 
accurate knowledge of numbers was originated, so also 
among the Egyptians geometry was invented for the 
reason above stated. 

" Thales first went to Egypt and thence introduced 
this study into GreeceA' 

(7) 



8 ARCHIMEDES 

^— '• 

But it is clear that the geometry of the Egyptians was 
almost entirely practical and did not go beyond the 
requirements of the land-surveyor, farmer or merchant 
They did indeed know, as far back as 2000 B.C., that in a 
triangle which has its sides proportional to 3, 4, 5 the 
angle contained by the two smaller sides is a right angle, 
and they used such a triangle as a practical means of 
drawing right angles. They had formulae, more or less 
inaccurate, for certain measurements, e.g. for the areas of 
certain triangles, parallel-trapezia, and circles. They had, 
further, in their construction of pyramids, to use the 
notion of similar right-angled triangles ; they even had a 
name, se-qet^ for the ratio of the half of the side of the 
base to the height, that is, for what we should call the 
co-tangent of the angle of slope. But not a single general 
theorem in geometry can be traced to the Egyptians. 
Their knowledge that the triangle (3, 4, 5) is right 
angled is far from implying any knowledge of the general 
proposition (Eucl. I., 47) known by the name of Pytha- 
goras. The science of geometry, in fact, remained to be 
discovered ; and this required the genius for pure specula- 
tion which the Greeks possessed in the largest measure 
anaong all the nations of the world 3 

UThales, who had travelled in Egypt and there learnt 
what the priests could teach him on the subject, intro- 
duced geometry into GreecCj^ Almost the whole of 
Greek science and philosophy begins with Thales. His 
date was about 624-547 B.C. First of the Ionian 
philosophers, and declared one of the Seven Wise Men 
in 582-581, he shone in all fields, as astronomer, mathe- 
matician, engineer, statesman and man of business. In 
astronomy he predicted the solar eclipse of 28 May, 585, 
discovered the inequality of the four astronomical seasons, 
and counselled the use of the Little Bear instead of the 
Great Bear as a means of finding the pole. In geometry 
the following theorems are attributed to him — and their 
character shows how the Greeks had to begin at the very 



GREEK GEOMETRY TO ARCHIMEDES 9 

beginning of the theory — (i) that a circle is bisected by 
any diameter (Euc). I., Def. 17), (2) that the angles at 
the base of an isosceles triangle are equal (Eucl. I., 5), 
(3) that, if two straight lines cut one another, the 
vertically opposite angles are equal (Eucl. I., 15), (4) 
that, if two triangles have two angles and one side 
respectively equal, the triangles are equal in all respects 
(Eucl. I., 26). He is said (5) to have been the first to 
inscribe a right-angled triangle in a circle : which must 
mean that he was the first to discover that the angle in a 
semicircle is a right angle; He also solved two problems 
in practical geometry: (i) he showed how to measure 
the distance from the land of a ship at sea (for this he is 
said to have used the proposition numbered (4) above), 
and (2) he measured the heights of pyramids by means 
of the shadow thrown on the ground (this implies the use 
of similar triangles in the way that the Egyptians had 
usedthem in the construction of pyramids). 

(Alter Thales come the Pythagoreans. We are told 
that the Pythagoreans were the first to use the term 
/jLadrjfMara (literally "subjects of instruction") in the 
specialised sense of " mathematics " ; they, too, first 
advanced mathematics as a study pursued for its 
own sake and made it a part of a liberal education. ' 
Pythagoras, son of Mnesarchus, was born in Samos 
about 572 B.C., and died at a great age (75 or 80) at 
Metapontum. His interests were as various as those of 
Thales ; his travels, all undertaken in pursuit of know- 
ledge, were probably even more extended. Like Thales, 
and perhaps at his suggestion, he visited Egypt and 
studied there for a long period (22 years, some say) J 

It is difficult to disentangle from the body of 
Pythagorean doctrines the portions which are due to 
Pythagoras himself because of the habit which the 
members of the school had of attributing everything to 
the Master {avro^ ecpa, ipse dixit). In astronomy two 
things at least may safely be attributed to him ; he held 



10 ARCHIMEDES 

that the earth is spherical in shape, and he recognised 
that the sun, moon and planets have an independent 
motion of their own in a direction contrary to that of 
the daily rotation ; he seems, however, to have adhered 
to the geocentric view of the universe, and it was his 
successors who evolved the theory that the earth does not 
remain at the centre but revolves, like the other planets 
and the sun and moon, about the " central fire ". Per- 
haps his most remarkable discovery was the dependence 
of the musical intervals on the lengths of vibrating 
strings, the proportion for the octave being 2 : i, for the 
fifth 3 : 2 and for the fourth 4:3. In arithmetic he 
was the first to expound the theory of means and of pro- 
portion as applied to commensurable quantities. He laid 
the foundation of the theory of numbers by considering 
the properties of numbers as such, namely, prime 
numbers, odd and even numbers, etc. By means of 
figured numbers, square, oblong, triangular, etc. (repre- 
sented by dots arranged in the form of the various 
figures) he showed the connexion between numbers and 
geometry. In view of all these properties of numbers, 
we can easily understand how the Pythagoreans came 
to " liken all things to numbers " and to find in the 
principles of numbers the principles of all things ("all 
things are numbers " ). 

We come now to Pythagoras's achievements in 
geometry. There is a story that, when he came home 
from Egypt and tried to found a school at Samos, he 
found the Samians indifferent, so that he had to take 
special measures to ensure that his geometry might not 
perish with him. Going to the gymnasium, he sought 
out a well-favoured youth who seemed likely to suit his 
purpose, and was withal poor, and bribed him to learn 
geometry by promising him sixpence for every proposition 
that he mastered. Very soon the youth got fascinated 
by the subject for its own sake, and Pythagoras rightly 
judged that he would gladly go on without the sixpence. 



GREEK GEOMETRY TO ARCHIMEDES ii 

He hinted, therefore, that he himself was poor and must 
try to earn his living instead of doing mathematics ; 
whereupon the youth, rather than give up the study, 
volunteered to pay sixpence to Pythagoras for each 
proposition. 

fin geometry Pythagoras set himself to lay the founda- 
tions of the subject, beginning with certain important 
definitions and investigating the fundamental principles. 
Of propositions attributed to him the most famous is, of 
course, the theorem that in a right-angled triangle the 
square on the hypotenuse is equal to the sum of the 
squares on the sides about the right angie**(Eucl. I., 
47) ; and, seeing that Greek tradition universally credits 
him with the proof of this theorem, we prefer to believe 
that tradition is right. This is to some extent confirmed 
by another tradition that Pythagoras discovered a general 
formula for finding two numbers such that the sum of 
their squares is a square number. This depends on the 
theory of the gnomon, which at first had an arithmetical 
signification corresponding to the geometrical use of it 
in Euclid, Book H. A figure in the shape of a gnomon 
put round two sides of a square makes it into a larger 
square. Now consider the number i represented by a 
dot. Round this place three other dots so that the four 
dots form a square (i 4- 3 = 2^). Round the four dots 
(on two adjacent sides of the square) place five dots at 
regular and equal distances, and we have another square 
(1 + 3 + 5 = 3^); and so on. The successive odd numbers 
I, 3, 5 . . „ were c^Xi^^ gnomons , and the general formula is 

I + 3 + 5 + • • • + (2« - I) = «^. 
Add the next odd number, i.e. 2«+ i, and we have 
n^ + {2n + i) = (« + i)^. In order, then, to get two 
square numbers such that their sum is a square we have 
only to see that 2w + i is a square. Suppose that 2n + i 
= fi^ \ then n = \{m^ - i), and we have {\{m^ - i)}^ + n^ 
= {i(w^+ i)p, where m is any odd number; and this is 
the general formula attributed to Pythagoras. 



12 ARCHIMEDES 

Proclus also attributes to Pythagoras the theory of 
proportionals and the construction of the five " cosmic 
figures," the five regular solids. 

One of the said solids, the dodecahedron, has twelve 
pentagonal faces, and the construction of a regular penta- 
gon involves the cutting of a straight line ** in extreme 
and mean ratio" (Eucl. II., ii, and VI., 30), which is a 
particular case of the method known as the application of 
areas. How much of this was due to Pythagoras him- 
self we do not know ; but the whole method was at all 
events fully worked out by the Pythagoreans and proved 
one of the most powerful of geometrical methods. The 
most elementary case appears in Euclid, I., 44, 45, where 
it is shown how to apply to a given straight line as base 
a parallelogram, having a given angle (say a rectangle) 
and equal in area to any rectilineal figure ; this construc- 
tion is the geometrical equivalent of arithmetical division. 
The general case is that in which the parallelogram, 
though applied to the straight line, overlaps it or falls 
short of it in such a way that the part of the parallelo- 
gram which extends beyond, or falls short of, the paral- 
lelogram of the same angle and breadth on the given 
straight line itself (exactly) as base is similar to another 
given parallelogram (Eucl. VI., 28, 29). This is the 
geometrical equivalent of the most general form of 
quadratic equation ax ± m^ = C, so far as it has real 
roots ; while the condition that the roots may be real 
was also worked out ( = Eucl. VI., 27). It is important 
to note that this method of application of areas was 
directly used by Apollonius of Perga in formulating the 
fundamental properties of the three conic sections, which 
properties correspond to the equations of the conies in 
Cartesian co-ordinates ; and the names given by Apollo- 
nius (for the first time) to the respective conies are taken 
from the theory, parabola {irapafioXri) meaning " applica- 
tion " (i.e. in this case the parallelogram is applied to the 
straight line exactly), hyperbola {yirepfiokri), " exceeding " 



GREEK GEOMETRY TO ARCHIMEDES 13 

(i.e. in this case the parallelogram exceeds or overlaps 
the straight line), ellipse (eXkec^L^), ''falling short" (i.e. 
the parallelogram falls short of the straight line). 

Another problem solved by the Pythagoreans is that 
of drawing a rectilineal figure equal in area to one given 
rectilineal figure and similar to another. Plutarch 
mentions a doubt as to whether it was this problem or 
the proposition of Euclid I., 47, on the strength of which 
Pythagoras was said to have sacrificed an ox. 

The main particular applications of the theorem of 
the square on the hypotenuse (e.g. those in Euclid, Book 
II.) were also Pythagorean ; the construction of a square 
equal to a given rectangle (Eucl. II., 14) is one of them 
and corresponds to the solution of the pure quadratic 
equation ;r^ = ad. 

The Pythagoreans proved the theorem that the sum 
of the angles of any triangle is equal to two right 
angles (Eucl. I., 32). 

Speaking generally, we may say that the Pythagorean 
geometry covered the bulk of the subject-matter of 
Books L, II., IV., and VI. of Euclid (with the qualifica- 
tion, as regards Book VI., that the Pythagorean theory 
of proportion applied only to commensurable magnitudes). 
Our information about the origin ot the propositions of 
Euclid, Book III., is not so complete; but it is certain 
that the most important of them were well known to 
Hippocrates of Chios (vv^ho flourished in the second half of 
the fifth century, and lived perl^aps from about 470 to 
400 B.C.), whence we conclude that the main propositions 
of Book III. were also included in the Pythagorean 
geometry. 

Lastly, the Pythagoreans discovered the existence of 
incommensurable lines, or of irrationals. This was, 
doubtless, first discovered with reference to the diagonal 
of a square which is incommensurable with the side, 
being in the ratio to it of ^2 to i. The Pythagorean 
proot of this particular case survives in Aristotle and in 



t4 ARCHIMEDES 

a proposition interpolated in Euclid's Book X. ; it is by 
a reductio ad absurdum proving that, if the diagonal is 
commensurable with the side, the same number must be 
both odd and even. This discovery of the incommen- 
surable was bound to cause geometers a great shock, 
because it showed that the theory of proportion invented 
by Pythagoras was not of universal application, and 
therefore that propositions proved by means of it were 
not really established. Hence the stories that the dis- 
covery of the irrational was for a time kept secret, and 
that the first person who divulged it perished by shipwreck. 
The fatal flaw thus revealed in the body of geometry 
was not removed till Eudoxus (408-355 B.C.) discovered 
the great theory of proportion (expounded in Euclid's 
Book v.), which is applicable to incommensurable as 
well as to commensurable magnitudes. 

By the time of Hippocrates of Chios the scope of 
Greek geometry was no longer even limited to the Ele- 
ments ; certain special problems were also attacked 
which were beyond the power of the geometry of the 
straight line and circle, and which were destined to play 
a great part in determining the direction taken by Greek 
geometry in its highest flights. The main problems in 
question were three : (i) the doubling of the cube, (2) the 
trisection of any angle, (3) the squaring of the circle ; 
and from the time of Hippocrates onwards the investiga- 
tion of these problems proceeded pari passu with the 
completion of the body of the Elements. 

Hippocrates himself is an example of the concurrent 
study of the two departments. On the one hand, he 
was the first of the Greeks who is known to have com- 
piled a book of Elements. This book, we may be sure, 
contained in particular the most important propositions 
about the circle included in Euclid, Book IH. But a 
much more important proposition is attributed to Hip- 
pocrates ; he is said to have been the first to prove that 
circles are to one another as the squares on their dia- 



GREEK GEOMETRY TO ARCHIMEDES 15 

meters ( = Eucl. XII., 2), with the deduction that similar 
segments of circles are to one another as the squares on 
their bases. These propositions were used by him in 
his tract on the squaring of lu7ies^ which was intended to 
lead up to the squaring of the circle. The latter problem 
is one which must have exercised practical geometers 
from time immemorial. Anaxagoras for instance (about 
500-428 B.C.) is said to have worked at the problem 
while in prison. The essential portions of Hippocrates's 
tract are preserved in a passage of Simplicius (on Aris- 
totle's Physics), which contains substantial fragments 
from Eudemus's History of Geometry. Hippocrates 
showed how to square three particular lunes of different 
forms, and then, lastly, he squared the sum of a certain 
circle and a certain lune. Unfortunately, however, the 
last-mentioned lune was not one of those which can be 
squared, and so the attempt to square the circle in this 
way failed after all. 

Hippocrates also attacked the problem of doubling 
the cube. There are two versions of the origin of this 
famous problem. According to one of them, an old 
tragic poet represented Minos as having been dissatisfied 
with the size of a tomb erected for his son Glaucus, and 
having told the architect to make it double the size, re- 
taining, however, the cubical form. According to the 
other, the Delians, suffering from a pestilence, were told 
by the oracle to double a certain cubical altar as a means 
of staying the plague. Hippocrates did not, indeed, 
solve the problem, but he succeeded in reducing it to 
another, namely, the problem of finding two mean pro- 
portionals in continued proportion between two given 
straight lines, i.e. finding x^y such that a\x = x'.y = 
y : b, where a, b are the two given straight lines. It is 
easy to see that, \i a \ x = x \ y = y \b, then bla = (xjaf, 
and, as a particular case, if ^ = 2^, x^ = 2^^ so that the 
side of the cube which is double of the cube of side a 
is found. 



i6 ARCHIMEDES 

The problem of doubling the cube was henceforth 
tried exclusively in the form of the probiem of the two 
mean proportionals. Two significant early solutions are 
on record. 

( I ) Archy tas of Tarentum (who flourished in first half of 
fourth century B.C.) found the two mean proportionals by 
a very striking construction in three dimensions, which 
shows that solid geometry, in the hands of Archytas at 
least, was already well advanced. The construction was 
usually called mechanical, which it no doubt was in form, 
though in reality it was in the highest degree theoretical. 
It consisted in determining a point in space as the inter- 
section of three surfaces : (a) a cylinder, (J?) a cone, (c) 
an " anchor-ring " with internal radius = o. (2) Mensech- 
mus, a pupil of Eudoxus, and a contemporary of Plato, 
found the two mean proportionals by means of conic 
sections, in two ways, (a) by the intersection of two para- 
bolas, the equations of which in Cartesian co-ordinates 
would be x^ = ay,)r = bx, and (yS) by the intersection 
of a parabola and a rectangular hyperbola, the corre- 
sponding equations being x^ = ay, and xy = ab respec- 
tively. It would appear that it was in the effort to solve 
this problem that Menaechmus discovered the conic 
sections, which are called, in an epigram by Eratosthenes, 
" the triads of Menaechmus ". 

The trisection of an angle was effected by means of a 
curve discovered by Hippias of Elis, the sophist, a con- 
temporary of Hippocrates as well as of Democritus and 
Socrates (470-399 B.C.). TJie curve was called the 
quadratrix because it also served (in the hands, as we 
are told, of Dinostratus, brother of Menaechmus, and of 
Nicomedes) for squaring the circle. It was theoretically 
constructed as the locus of the point of intersection of two 
straight lines moving at uniform speeds and in the same 
time, one motion being angular and the other rectilinear. 
Suppose OA, OB are two radii of a circle at right angles 
to one another. Tangents to the circle at A and B, 



GREEK GEOMETRY TO ARCHIMEDES i; 

meeting at C, form with the two radii the square OACB. 
The radius OA is made to move uniformly about O, the 
centre, so as to describe the angle AOB in a certain 
time. Simultaneously AC moves parallel to itself at 
uniform speed such that A just describes the line AO 
in the same length of time. The intersection of the 
moving radius and AC in their various positions traces 
out the quadratrix. 

The rest of the geometry which concerns us was mostly 
th3 work of a few men, Democritus of Abdera, Theodorus 
of Cyrene (the mathematical teacher of Plato), Theaetetus, 
Eudoxus, and Euclid. The actual writers of Elements 
of whom we hear were the following. Leon, a little 
younger than Eudoxus (408-355 B.C.), was the author 
of a collection of propositions more numerous and 
more serviceable than those collected by Hippocrates. 
Theudius of Magnesia, a contemporary of Menaech- 
mus and Dinostratus, "put together the elements ad- 
mirably, making many partial or limited propositions 
more general ". Theudius' s book was no doubt the 
geometrical text-book of the Academy and that used by 
Aristotle. 

Theodorus of Cyrene and Theaetetus generalised the 
theory of irrationals, and we may safely conclude that a 
great part of the substance of Euclid's Book X. (on 
irrationals) was due to Theaetetus. Theaetetus also wrote 
on the five regular solids (the tetrahedron, cube, octa- 
hedron, dodecahedron, and icosahedron), and Euclid 
was therefore no doubt equally indebted to Theaetetus 
for the contents of his Book XIII. In the matter of 
Book XII. Eudoxus was the pioneer. These facts are 
confirmed by the remark of Proclus that Euclid, in com- 
piling his Elements, collected many of the theorems of 
Eudoxus, perfected many others by Theaetetus, and 
brought to irrefragable demonstration the propositions 
which had only been somewhat loosely proved by his pre- 
decessors, 



l8 ARCHIMEDES 

Eudoxus (about 408-355 B.C.) was perhaps the greatest 
of all Archimedes's predecessors, and it is his achieve- 
ments, especially the discovery of the method of exhaus- 
tion, which interest us in connexion with Archimedes. 

In astronomy Eudoxus is famous for the beautiful 
theory of concentric spheres which he invented to explain 
the apparent motions of the planets, and, particularly, 
their apparent stationary points and retrogradations. 
The theory applied also to the sun and moon, for which 
Eudoxus required only three spheres in each case. He 
represented the motion of each planet as compounded 
of the rotations of four interconnected spheres about 
diameters, all of which pass through the centre of the 
earth. The outermost sphere represents the daily rota- 
tion, the second a motion along the zodiac circle or 
ecliptic ; the poles of the third sphere, about which that 
sphere revolves, are fixed at two opposite points on the 
zodiac circle, and are carried round in the motion of the 
second sphere ; and on the surface of the third sphere 
the poles of the fourth sphere are fixed ; the fourth 
sphere, revolving about the diameter joining its two 
poles, carries the planet which is fixed at a point on its 
equator. The poles and the speeds and directions of 
rotation are so chosen that the planet actually describes 
a hippopede, or horse-fetter, as it was called (i.e. a figure of 
eight), which lies along and is longitudinally bisected by 
the zodiac circle, and is carried round that circle. As 
a tour de force of geometrical imagination it would be 
difficult to parallel this hypothesis. 

In geometry Eudoxus discovered the great theory of 
proportion, applicable to incommensurable as well as com- 
mensurable magnitudes, which is expounded in Euclid, 
Book v., and which still holds its own and will do so for 
all time. He also solved the problem of the two mean 
proportionals by means of certain curves, the nature of 
which, in the absence of any description of them in our 
sources, can only be conjectured. 



GREEK GEOMETRY TO ARCHIMEDES 19 

Last of all, and most important for our purpose, is his 
use of the famous method of exhaustion for the measure- 
ment of the areas of curves and the volumes of solids. 
The example of this method which will be most familiar 
to the reader is the proof in Euclid XIL, 2, of the theorem 
that the areas of circles are to one another as the squares 
on their diameters. The proof in this and in all cases 
depends on a lemma which forms Prop, i of Euclid's 
Book X. to the effect that, if there are two unequal 
magnitudes of the same kind and from the greater you 
subtract not less than its half, then from the remainder 
not less than its half, and so on continually, you will at 
length have remaining a magnitude less than the lesser 
of the two magnitudes set out, however small it is. 
Archimedes says that the theorem of Euclid XIL, 2, was 
proved by means of a certain lemma to the effect that, if 
we have two unequal magnitudes (i.e. lines, surfaces, or 
solids respectively), the greater exceeds the lesser by 
such a magnitude as is capable, if added continually to 
itself, of exceeding any magnitude of the same kind as 
the original magnitudes. This assumption is known as 
the Axiom or Postulate of Archimedes, though, as he 
states, it was assumed before his time by those who 
used the method of exhaustion. It is in reality used 
in Euclid's lemma (Eucl. X., i) on which Euclid 
XIL, 2, depends, and only differs in statement from 
Def 4 of Euclid, Book V., which is no doubt due to 
Eudoxus. 

The method of exhaustion was not discovered all at 
once ; we find traces of gropings after such a method 
before it was actually evolved. It was perhaps Antiphon, 
the sophist, of Athens, a contemporary of Socrates (470- 
399 B.C.), who took the first step. He inscribed a square 
(or, according to another account, an equilateral triangle) 
in a circle, then bisected the arcs subtended by the sides, 
and so inscribed a polygon of double the number of 
sides ; he then repeated the process, and maintained that, 



20 ARCHIMEDES 

by continuing it, we should at last arrive at a polygon 
with sides so small as to make the polygon coincident, 
with the circle. Though this was formally incorrect, it 
nevertheless contained the germ of the method of ex- 
haustion. 

Hippocrates, as we have seen, is said to have proved 
the theorem that circles are to one another as the squares 
on their diameters, and it is difficult to see how he could 
have done this except by some form, or anticipation, of 
the method. There is, however, no doubt about the 
part taken by Eudoxus ; he not only based the method 
on rigorous demonstration by means of the lemma or 
lemmas aforesaid, but he actually applied the method to 
find the volumes (i) of any pyramid, (2) of the cone, 
proving (i) that any pyramid is one third part of the 
prism which has the same base and equal height, and (2) 
that any cone is one third part of the cylinder which has 
the same base and equal height. Archimedes, however, 
tells us the remarkable fact that these two theorems 
were first discovered by Democritus (who flourished 
towards the end of the fifth century B.C.), though he was 
not able to prove them (which no doubt means, not that 
he gave no sort of proof, but that he was not able 
to establish the propositions by the rigorous method 
of E'ldoxus). Archimedes adds that we must give no 
small share of the credit for these theorems to Demo- 
critus ; and this is another testimony to the marvellous 
powers, in mathematics as well as in other subjects, 
of the great man who, in the words of Aristotle, 
"seems to have thought of everything". We know 
from other sources that Democritus wrote on irrationals ; 
he is also said to have discussed the question of two 
parallel sections of a cone (which were evidently sup- 
posed to be indefinitely close together), asking whether 
we are to regard them as unequal or equal : " for if 
they are unequal they will make the cone irregular as 
having many indentations, like steps, and unevennesses, 



GREEK GEOMETRY TO ARCHIMEDES 21 

but, if they are equal, the cone will appear to have the 
property of the cylinder and to be made up of equal, 
not unequal, circles, which is very absurd ". This ex- 
planation shows that Democritus was already close on 
the track of infinitesimals. 

Archimedes says further that the theorem that spheres 
are in the triplicate ratio of their diameters was proved 
by means of the same lemma, The proofs of the proposi- 
tions about the volumes of pyramids, cones and spheres 
are, of course, contained in Euclid, Book XI I. (Props. 
3-7 Cor., 10, 16-18 respectively). 

It is no doubt desirable to illustrate Eudoxus's 
method by one example. We will take one of the sim- 
plest, the proposition (Eucl. XI I. , 10) about the cone. 
Given ABCD, the circular base of the cylinder which 
has the same base as the cone and equal height, we in- 
scribe the square ABCD ; we then bisect the arcs sub- 
tended by the sides, and draw the regular inscribed 
polygon of eight sides, then similarly we draw the 
regular inscribed polygon of sixteen sides, and so on. 
We erect on each regular polygon the prism which has 
the polygon for base, thereby obtaining successive prisms 
inscribed in the cylinder, and of the same height with it. 
Each time we double the number of sides in the base of 
the prism we take away more than half of the volume 
by which the cylinder exceeds the prism (since we take 
away more than half of the excess of the area of the 
circular base over that of the inscribed polygon, as in 
Euclid XII., 2). Suppose now that V is the volume 
of the cone, C that of the cylinder. We have to prove 
that C = 3V. If C is not equal to 3V, it is either 
greater or less than 3V. 

Suppose (I) that C> 3V, and that C = 3V + E. 
Continue the construction of prisms inscribed in the 
cylinder until the parts of the cylinder left over outside 
the final prism (of volume P) are together less than E. 



22 ARCHIMEDES 

Then C - P<E. 

But C - 3V = E ; 

•' Therefore P > 3 V. 

But it has been proved in earlier propositions that P is 
equal to three times the pyramid with the same base as 
the prism and equal height. 

Therefore that pyramid is greater than V, the volume 
of the cone : which is impossible, since the cone encloses 
the pyramid. 

Therefore C is not greater than 3V. 

Next (2) suppose that C < 3V, so that, inversely, 

V>-C. 

3 

This time we inscribe successive pyramids in the cone 
until we arrive at a pyramid such that the portions of 
the cone left over outside it are together less than the 

excess of V over - C. It follows that the pyramid is 

greater than - C. Hence the prism on the same base as 

the pyramid and inscribed in the cylinder (which prism is 
three times the pyramid) is greater than C : which is 
impossible, since the prism is enclosed by the cylinder, 
and is therefore less than it. 

Therefore V is not greater than - C, or C is not less 

than 3 V. 

Accordingly C, being neither greater nor less than 3 V, 

must be equal to it ; that is, V = - C. 

It only remains to add that Archimedes is fully ac- 
quainted with the main properties of the conic sections. 
These had already been proved in earlier treatises, which 
Archimedes refers to as the ** Elements of Conies ". We 
know of two such treatises, (i) Euclid's four Books on 



GREEK GEOMETRY TO ARCHIMEDES 23 

Conies, (2) a work by one Aristaeus called " Solid Loci," 
probably a treatise on conies regarded as loci. Both 
these treatises are lost ; the former was, of course, super- 
seded by Apollonius's great work on Conies in eight 
Books. 



CHAPTER III. 

THE WORKS OF ARCHIMEDES. 

The range of Archimedes's writings will be gathered 
from the list of his various treatises. An extraordinarily 
large proportion of their contents represents entirely new 
discovieries of his own. 'jHe was no compiler or writer of 
text-books, and in this respect he differs from Euclid and 
Apollonius, whose work largely consisted in systematis- 
ing and generalising the methods used and the results 
obtained by earlier geometers. There is in Archimedes 
no mere working-up of existing material ; his objective 
is always something new, some definite addition to the 
sum of knowledge. J Confirmation of this is found in the 
introductory letters prefixed to most of his treatises. In 
them we see the directness, simplicity and humanity of 
the man. There is full and generous recognition of the 
work of predecessors and contemporaries ; his estimate 
of the relation of his own discoveries to theirs is obviously 
just and free from any shade of egoism. Hlis manner is 
to state what particular discoveries made by his prede- 
cessors had suggested to him the possibility of extending 
them in new directions ; thus he says that, in connexion 
with the efforts of earlier geometers to square the circle, 
it occurred to him that no one had tried to square a para- 
bolic segment ; he accordingly attempted the problem 
and finally solved it. Similarly he describes his dis- 
coveries about the volumes and surfaces of spheres and 
cylinders as supplementing the theorems of Eudoxus 

(24) 



THE WORKS OF ARCHIMEDES 25 

about the pyramid, the cone and the cylinderJ He does 
not hesitate to say that certain problems baffled him for 
a long time ; in one place he positively insists, for the 
purpose of pointing a moral, on specifying two propo- 
sitions which he had enunciated but which on further in- 
vestigation proved to be wrong. 

The ordinary MSS. of the Greek text of Archimedes 
give his works in the following order : — 

1. On the Sphere and Cylinder (two books). 

2. Measurement of a Circle. 

3. On Conoids and Spheroids. 

4. On Spirals. 

5. On Plane Equilibriums (two books). 

6. The Sandreckoner. 

7. Quadrature of a Parabola. 

A most important addition to this list has been made 
in recent years through an extraordinary piece of good 
fortune. In 1906 J. L. Heiberg, the most recent editor 
of the text of Archimedes, discovered a palimpsest of 
mathematical content in the "Jerusalemic Library" of 
one Papadopoulos Kerameus at Constantinople. This 
proved to contain writings of Archimedes copied in 
a good hand of the tenth century. An attempt had been 
made (fortunately with only partial success) to wash out 
the old writing, and then the parchment was used again 
to write a Euchologion upon. However, on most of the 
leaves the earlier writing remains more or less legible. 
The important fact about the MS. is that it contains, 
besides substantial portions of the treatises previously 
known, (i) a considerable portion of the work, in two 
books. On Floating Bodies^ which was formerly supposed 
to have been lost in Greek and only to have survived in 
the translation by Wilhelm of Morbeke, and (2) most 
precious of all, the greater part of the book called The 
Method^ treating of Mechanical Problems and addressed 
to Eratosthenes. The important treatise so happily 
recovered is now included in Heiberg's new (second) 



y 



26 ARCHIMEDES 

edition of the Greek text of Archimedes (Teubner, 
1 91 0-15), and some account of it will be given in the 
n ext c hapter. 

Tlhe order in which the treatises appear in the MSS. 
was not the order of composition ; but from *he various 
prefaces and from internal evidence generally we are able 
to establish the following as being approximately the 
chronological sequence : — 

1. On Plane Equilibriums, I. 

2. Quadrature of a Parabola. 

3. On Plane Equilibriums, II. 

4. The Method. 

5. On the Sphere and Cylinder, I, II. 

6. On Spirals. 

7. On Conoids and Spheroids. 

8. On Floating Bodies, I, II. 

9. Measurement of a Circle. 
10. The Sandreckoner. 

In addition to the above we have a collection of 
geometrical propositions which has reached us through 
the Arabic with the title ''Liber assumptorum Archi- 
medis". They were not written by Archimedes in their 
present form, but were probably collected by some later 
Greek writer for the purpose of illustrating some ancient 
work. It is, however, quite likely that some of the pro- 
positions, which are remarkably elegant, were of Archi- 
medean origin, notably those concerning the geometrical 
figures made with three and four semicircles respectively 
and called (from their shape) (i) the shoemaker s knife 
and (2) the Salinon or salt-cellar, and another theorem 
which bears on the trisection of an angle. 

An interesting fact which we now know from Arabian 
sources is that the formula for the area of any triangle 
in terms of its sides which we write in the form 

A = ^\s{s - a){s - b){s - c)\, 
and which was supposed to be Heron's because Heron 
gives the geometrical proof of it, was really due to 
Archimedes. 



V 



THE WORKS OF ARCHIMEDES 2; 

Archimedes is further credited with the authorship of ' 
the famous Cattle-Problem enunciated in a Greek epi- . 
gram edited by Lessing in 1773. According to its^ 
heading the problem was communicated by Archimedes t 
to the mathematicians at Alexandria in a letter to** 
Eratosthenes ; and a scholium to Plato's Charmides -^ ^^ 
speaks of the problem " called by Archimedes the Cattle-"^ ^ " 
Problem ". It is an extraordinarily difficult problem in 
indeterminate analysis, the solution of which involves 
enormous figures. 

^ Of lost works of Archimedes the following can be . 

^identified : — v • ' '^^ • : * ^ :■ ^-K 

1. Investigations relating to polyhedra are referred to -^^; 
by Pappus, who, after speaking of the five regular solids, 
gives a description of thirteen other polyhedra discovered 
by Archimedes which are semi-regular, being contained 
by polygons equilateral and equiangular but not similar. 
OwQ. at least of these semi-regular solids was, however, 
already known to Plato. >> 

2. A book of arithmetical content entitled Principles^ - 
dealt, as we learn from Archimedes himself, with the 
naming of numbers, and expounded a system of express- 
ing large numbers which could not be written in the 
ordinary Greek notation. In setting out the same system x 
in the Sandreckoner (see Chapter V. below), Archimedes '^ 
explains that he does so for the benefit of those who had j 
not seen the earlier work. '~^ < 

3. On Balances (or perhaps levers). Pappus says that 
in this work Archimedes proved that *' greater circles 
overpower lesser circles when they rotate about the same 
centre ". 

4. A book On Centres of Gravity is alluded to by Sim- 
plicius. It is not, however, certain that this and the 
last-mentioned work were separate treatises. Possibly 
Book I. On Plane Equilibriums may have been part of a 
larger work (called perhaps Elements of Mechanics), and 
On Balances may have been an alternative title. The 

J 



28 ARCHIMEDES 

title On Centres of Gravity may be a loose way of referring 
to the same treatise. 

5. Catoptrica^ an optical work from which Theon of 
Alexandria quotes a remark about refraction. 

6. On Sphere-makings a mechanical work on the con- 
struction of a sphere to represent the motions of the 
heavenly bodies (cf. pp. 5-6 above). 

Arabian writers attribute yet further works to Archi- 
medes, (i) On the circle, (2) On a heptagon in a circle, 
(3) On circles touching one another, (4) On parallel lines, 
(5) On triangles, (6) On the properties of right-angled 
triangles, (7) a book of Data ; but we have no con- 
firmation of these statements. 



CHAPTER IV. 

GEOMETRY IN ARCHIMEDES. 

The famous French geometer, Chasles, drew an instruc- 
tive distinction between the predominant features of the 
geometry of the two great successors of Euclid, namely, 
Archimedes and Apollonius of Perga (the "great geo- 
meter," and author of the classical treatise on Conies). 
The works of these two men may, says Chasles, be 
regarded as the origin and basis of two great inquiries 
which seem to share between them the domain of 
geometry. Apollonius is concerned with the Geometry 
of Forms and Situations^ while in Archimedes we find 
the Geometry of Measurements, dealing with the quadrature 
of curvilinear plane figures and with the quadrature and 
cubature of curved surfaces, investigations which gave 
birth to the calculus of the infinite conceived and brought 
to perfection by Kepler, Cavalieri, Fermat, Leibniz and 
Newton. 

In geometry Archimedes stands, as it were, on the 
shoulders of Eudoxus in that he applied the method of 
exhaustion to new and more difficult cases of quadrature 
and cubature. Further, in his use of the method he 
introduced an interesting variation of the procedure as 
we know it from Euclid. Euclid (and presumably 
Eudoxus also) only used inscribed figures, *' exhausting " 
the figure to be measured, and had to invert the second 
half of the reductio ad absurdum to enable approximation 
from below (so to speak) to be applied in that case also. 

(29) 3 



30 ARCHIMEDES 

pTrchimedes, on the other hand, approximates from above 
as well as from below ; he approaches the area or volume 
to be measured by taking closer and closer circumscribed 
figures, as well as inscribed, and thereby compressing, as 
it were, the inscribed and circumscribed figure into one, 
so that they ultimately coincide with one another and 
with the figure to be measured. But he follows the 
cautious method to which the Greeks always adhered ; 
he never says that a given curve or surface is the limiting 
form of the inscribed or circumscribed figure ; all that he 
asserts is that we can approach the curve or surface as 
nearly as we please. 

The deductive form of proof by the method of exhaus- 
tion is apt to obscure not only the way in which the 
results were arrived at but also the real character of the 
procedure followed. What Archimedes actually does in 
certain cases is to perform what are seen, when the 
analytical equivalents are set down, to be Y&2i\ integrations; j 
this remark applies to his investigation of the areas of a 
parabolic segment and a spiral respectively, the surface 
and volume respectively of a sphere and a segment of a 
sphere, and the volume of any segments of the solids of I 
revolution of the second degree. The result is, as a rule, ' 
only obtained after a long series of preliminary proposi- 
tions, all of which are links in a chain of argument 
elaborately forged for the one purpose. The method 
suggests the tactics of some master of strategy who fore- 
sees everything, eliminates everything not immediately 
conducive to the execution of his plan, masters every 
position in its order, and then suddenly (when the very 
elaboration of the scheme has almost obscured, in the 
mind of the onlooker, its ultimate object) strikes the final 
blow. Thus we read in Archimedes proposition after 
proposition the bearing of which is not immediately 
obvious but which we find infallibly used later on ; and 
we are led on by such easy stages that the difficulty of 
the original problem, as presented at the outset, is 



GEOMETRY IN ARCHIMEDES 31 

scarcely appreciated. As Plutarch says, " It is not 
possible to find in geometry more difficult and trouble- 
some questions, or more simple and lucid explanations ". 
But it is decidedly a rhetorical exaggeration when Plutarch 
goes on to say that we are deceived by the easiness of 
the successive steps into the belief that any one could 
have discovered them for himself. On the contrary, the 
studied simplicity and the perfect finish of the treatises in- 
volve at the same time an element of mystery. Although 
each step depends upon the preceding ones, we are left 
in the dark as to how they were suggested to Archimedes. 
There is, in fact, much truth in a remark of Wallis to the 
effect that he seems "as it were of set purpose to have 
covered up the traces of his investigation as if he had 
grudged posterity the secret of his method of inquiry 
while he wished to extort from them assent to his 
results ". 

A partial exception is now furnished by the Method ; 
for here we have (as it were) a lifting of the veil and a 
glimpse of the interior of Archimedes's workshop. He 
tells us how he discovered certain theorems in quadrature : 
and cubature, and he is at the same time careful to insist S 
on the difference between (i) the means which may serve 
to suggest the truth of theorems, although not furnishing - 
scientific proofs of them, and (2) the rigorous demon- 
strations of them by approved geometrical methods 
which must follow before they can be finally accepted as ^^ 
established. 2^^-' 

Writing to Eratosthenes he says: "Seeing in you, as *^ 

I say, an earnest student, a man of considerable eminence 

in philosophy and an admirer of mathematical inquiry 
when it comes your way, I have thought fit to write out 
for you and explain in detail in the same book the 
peculiarity of a certain method, which, when you see it, 
will put you in possession of a means whereby you can 
investigate some of the problems of mathematics by 
mechanics. This procedure is, I am persuaded, no less 



32 ARCHIMEDES 

useful for the proofs of the actual theorems as well. For 
certain things which first became clear to me by a 
mechanical method had afterwards to be demonstrated 
by geometry, because their investigation by the said 
method did not furnish an actual demonstration. But it 
is of course easier, when we have previously acquired 
by the method some knowledge of the questions, to 
supply the proof than it is to find the proof without any 
previous knowledge. This is a reason why, in the case 
of the theorems the proof of which Eudoxus was the first 
to discover, namely, that the cone is a third part of the 
cylinder, and the pyramid a third part of the prism, 
having the same base and equal height, we should give 
no small share of the credit to Democritus, who was the 
first to assert this truth with regard to the said figures, 
though he did not prove it. I am myself in the position 
of having made the discovery of the theorem now to be 
published in the same way ^as I made my earlier 
discoveries ; and I thought it desirable now to write out 
and publish the method, partly because I have already 
spoken of it and I do not want to be thought to have 
uttered vain words, but partly also because I am 
persuaded that it will be of no little service to mathe- 
matics ; for I apprehend that some, either of my 
contemporaries or of my successors, will, by means of 
the method when once established, be able to discover 
other theorems in addition, which have not occurred to 
me. 

" First then I will set out the very first theorem which 
became known to me by means of mechanics, namely, 
that Any segment of a section of a right-angled cone [i.e. 
a parabola] is four-thirds of the triangle which has the same 
base and equal height ; and after this I will give each of 
the other theorems investigated by the same method. 
Then, at the end of the booky, I will give the geometrical 
proofs of the propositions." ''^^. 

The following description will, I hope, give an idea of 



GEOMETRY IN ARCHIMEDES 33 

the general features of the mechanical method employed 
by Archimedes. Suppose that X is the plane or solid 
figure the area or content of which is to be found. The 
method in the simplest case is to weigh infinitesimal 
elements of X against the corresponding elements of 
another figure, B say, being such a figure that its area or 
content and the position of its centre of gravity are 
already known. The diameter or axis of the figure X 
being drawn, the infinitesimal elements taken are parallel 
sections of X in general, but not always, at right angles 
to the axis or diameter, so that the centres of gravity of 
all the sections lie at one point or other of the axis or 
diameter and their weights can therefore be taken as 
acting at the several points of the diameter or axis. In 
the case of a plane figure the infinitesimal sections are 
spoken of as parallel straight lines and in the case of a 
solid figure as parallel planes, and the aggregate of the 
infinite number of sections is said to make up the whole 
figure X. (Although the sections are so spoken of as 
straight lines or planes, they are really indefinitely narrow 
plane strips or indefinitely thin laminae respectively.) 
The diameter or axis is produced in the direction away 
from the figure to be measured, and the diameter or axis 
as produced is imagined to be the bar or lever of a 
balance. The object is now to apply all the separate 
elements of X at one point on the lever, while the cor- 
responding elements of the known figure B operate at 
different points, namely, where they actually are in the 
first instance. Archimedes contrives, therefore, to move 
the elements of X away from their original position and 
to concentrate them at one point on the lever, such that 
each of the elements balances, about the point of sus- 
pension of the lever, the corresponding element of B 
acting at its centre of gravity. The elements of X and B 
respectively balance about the point of suspension in 
accordance with the property of the lever that the weights 
are inversely proportional to the distances from the 



34 ARCHIMEDES 

fulcrum or point of suspension. Now the centre of 
gravity of B as a whole is known, and it may then be 
supposed to act as one mass at its centre of gravity. 
(Archimedes assumes as known that the sum of the 
" moments," as we call them, of all the elements of the 
figure B, acting severally at the points where they actually 
are, is equal to the moment of the whole figure applied 
as one mass at one point, its centre of gravity.) More- 
over all the elements of X are concentrated at the one 
fixed point on the bar or lever. If this fixed point is H, 
and G is the centre of gravity of the figure B, while C is 
the point of suspension, 

X : B = CG : CH. 

Thus the area or content of X is found. 

Conversely, the method can be used to find the centre 
of gravity of X when its area or volume is known before- 
hand. In this case the elements of X, and X itself, have 
to be applied where they are, and the elements of the 
known figure or figures have to be applied at the one 
fixed point H on the other side of C, and since X, B and 
CH are known, the proportion 

B : X = CG : CH 
determines CG, where G is the centre of gravity of X. 

The mechanical method is used for finding (i) the area 
of any parabolic segment, (2) the volume of a sphere and 
a spheroid, (3) the volume of a segment of a sphere and 
the volume of a right segment of each of the three coni- 
coids of revolution, (4) the centre of gravity (a) of a 
hemisphere, (J?) of any segment of a sphere, {c) of any 
right segment of a spheroid and a paraboloid of revolu- 
tion, and (d) of a half-cylinder, or, in other words, of a 
semicircle. 

Archimedes then proceeds to find the volumes of two 
solid figures, which are the special subject of the treatise. 
The solids arise as follows : — 

(i) Given a cylinder inscribed in a rectangular parallel- 
epiped on a square base in such a way that the two 



GEOMETRY IN ARCHIMEDES 35 

bases of the cylinder are circles inscribed in the opposite 
square faces, suppose a plane drawn through one side 
ot the square containing one base of the cylinder and 
through the parallel diameter of the opposite base of the 
cylinder. The plane cuts off a solid with a surface re- 
sembling that of a horse's hoof Archimedes proves that 
the volume of the solid so cut off is one sixth part of the 
volume of the parallelepiped. 

(2) A cylinder is inscribed in a cube in such a way 
that the bases ot the cylinder are circles inscribed in two 
oppo ite square faces. Another cylinder is inscribed 
which is similarly related to another pair of opposite 
faces. The two cylinders include between them a solid 
with all its angles rounded off; and Archimedes proves 
that the volume of this solid is two-thirds of that of the 
cube. 

Having proved these facts by the mechanical method, 
Archimedes concluded the treatise with a rigorous geo- 
metrical proof of both propositions by the method of 
exhaustion. The MS. is unfortunately jomewhat muti- 
lated at the end, so that a certain amount of restoration 
is necessary. 

I shall now attempt to give a short account of the 
other treatises of Archimedes in the order in which they 
appear in the editions. The first is — 

On the Sphere and Cylinder. 

Book I. begins with a preface addressed to Dositheus 
(a pupil of Conon), which reminds him that on a former 
occasion he had communicated to him the treatise proving 
that any segment of a "section of a right-angled cone" 
(i.e. a parabola) is four-thirds of the triangle with the same 
base and height, and adds that he is now sending the 
proofs of certain theorems which he has since discovered, 
and which seem to him to be worthy of comparison with 
Eudoxus's propositions about the volumes of a pyramid 
and a cone. The theorems are (i) that the surface of a 



31? ARCHIMEDES 

sphere is equal to four times its greatest circle (i.e. what 
we call a "great circle" of the sphere); (2) that the sur- 
face of any segment of a sphere is equal to a circle with 
radius equal to the straight line drawn from the vertex 
of the segment to a point on the circle which is the base 
of the segment ; (3) that, if we have a cylinder circum- 
scribed to a sphere and with height equal to the diameter, 
then (a) the volume of the cylinder is i^ times that of 
the sphere and (d) the surface of the cylinder, including 
its bases, is i-^ times the surface of the sphere. 

Next come a few definitions, followed by certain As- 
sumptions, two of which are well known, namely : — 

1. Of all lines which have the same extremities the 
straight line is the least (this has been made the basis of 
an alternative definition of a straight line). 

2. Of unequal lines ^ unequal surfaces and unequal 
solids the greater exceeds the less by such a magnitude as, 
when {continually) added to itself can be made to exceed 
any assigned magnitude among those which are comparable 
[with it and] with one another (i.e. are of the same kind). 
This is the Postulate of Archimedes. 

He also assumes that, of pairs of lines (including broken 
lines) and pairs of surfaces, concave in the same direction 
and bounded by the same extremities, the outer is greater 
than the inner. These assumptions are fundamental to 
his investigation, which proceeds throughout by means 
of figures inscribed and circumscribed to the curved lines 
or surfaces that have to be measured. 

After some preliminary propositions Archimedes finds 
(Props. 13, 14) the area of the surfaces (i) of a right 
cylinder,'(2) of a right cone. Then, after quoting certain 
Euclidean propositi6ns""about cones and cylinders, he 
passes to the main business of the book, the measure- 
ment of the volume and surface of a sphere and a segment 
of a sphere. By circumscribing and inscribing to a great 
circle a regular polygon of an even number of sides and 
making it revolve about a diameter connecting two op- 



GEOMETRY IN ARCHIMEDES 37 

posite angular points he obtains solids of revolution 
greater and less respectively than the sphere. In a 
series of propositions he finds expressions for (a) the 
surfaces, {b) the volumes, of the figures so inscribed and 
circumscribed to the sphere. Next he proves (Prop. 32) 
that, if the inscribed and circumscribed polygons which, 
by their revolution, generate the figures are similar, the 
surfaces of the figures are in the duplicate ratio, and their 
volumes in the triplicate ratio, of their sides. Then he 
proves that the surfaces and volumes of the inscribed and 
circumscribed figures respectively are less and greater 
than the surface and volume respectively to which the 
main propositions declare the surface and volume of the 
sphere to be equal (Props. 25, 2y, 30, 31 Cor.). Ke 
has now all the material for applying the method of ex- 
haustion and so proves the main propositions about the 
surface and volume of the sphere. The rest of the book 
applies the same procedure to a segment of the sphere. 
Surfaces of revolution are inscribed and circumscribed to 
a segment less than a hemisphere, and the theorem about 
the surface of the segment is finally proved in Prop. 42. 
Prop. 43 deduces the surface of a segment greater than 
a hemisphere. Prop. 44 gives the volume of the sector 
of the sphere which includes any segment. 

xBdokll/' begins with the problem of finding a sphere 
equal m volume to a given cone or cylinder ; this 
requires the solution of the problem of the two mean pro- 
portionals, which is accordingly assumed. Prop. 2 de-. 
duces, by means of I., 44, an expression for the volume 
of a segment of a sphere, and Props. 3, 4 solve the im- 
portant problems of cutting a given sphere by a plane 
so that {a) the surfaces, {b) the volumes, of the segments 
may have to one another a given ratio. The solution of 
the second problem (Prop. 4) is difficult. Archimedes 
reduces it to the problem of dividing a straight line AB 
into two parts at a point M such that 

MB : (a given length) = (a given area) : AMI 



38 ARCHIMEDES 

The solution of this problem with a determination of the 
limits of possibility are given in a fragment by Archi- 
medes, discovered and preserved for us by Eutocius in 
his commentary on the book ; they are effected by means 
of the points ot intersection of two conies, a parabola and 
a rectangular hyperbola. Three problems of construc- 
tion follow, the first two of which are to construct a seg- 
ment of a sphere similar to one given segment, and 
having (a) its volume, (d) its surface, equal to that of 
another given segment of a sphere. The last two pro- 
positions are interesting. Prop. 8 proves that, if V, V 
be the volumes, and S, S' the surfaces, of two segments 
into which a sphere is divided by a plane, V and S be- 
longing to the greater segment, then 

S^ :S'2>V: r>S^si 

Prop. 9 proves that, of all segments of spheres which 
have equal surfaces, the hemisphere is the greatest in 
volume. 

T/ie Measurement of a Circle. 

This treatise, in the form in which it has come down 
to us, contains only three propositions ; the second, being 
an easy deduction from Props, i and 3, is out of place in 
so far as it uses the result of Prop. 3. 

In Prop. I Archimedes inscribes and circumscribes to 
a circle a series of successive regular polygons, beginning 
with a square, and continually doubling the number of 
sides ; he then proves in the orthodox manner by the 
method of exhaustion that the area of the circle is equal 
to that of a right-angled triangle, in which the perpen- 
dicular is equal to the radius, and the base equal to 
the circumference, of the circle. Prop. 3 is the 
famous proposition in which Archimedes finds by 
sheer calculation upper and lower arithmetical limits to 



GEOMETRY IN ARCHIMEDES 39 

the ratio of the circumference of a circle to its diameter, 
or what we call tt ; the result obtained is 3T>7r>3TT- 
Archimedes inscribes and circumscribes successive regu- 
lar polygons, beginning with hexagons, and doubling 
the number of sides continually, until he arrives at inscribed 
and circumscribed regular polygons with 96 sides ; seeing 
then that the length of the circumference of the circle is in- 
termediate between the perimeters of the two polygons, 
he calculates the two perimeters in terms of the diameter 
of the circle. His calculation is based on two close 
approximations (an upper and a lower) to the value of 
^3, that being the cotangent of the angle of 30°, from 
which he begins to work. He assumes as known that 

TT- < \/3 < -ToT . In the text, as we have it, only the 

results of the steps in the calculation are given, but they 
involve the finding of approximations to the square roots 
of several large numbers: thus 1172!^ is given as the ap- 
proximate value of N/(i373943ff), 30i3f as that of 
^(9082321) and 1838^ as that of ^(3380929). Inthis 

T A ^88 

way Archimedes arrives at -- — j as the ratio of the peri- 
meter of the circumscribed polygon of 96 sides to the di- 
ameter of the circle ; this is the figure which he rounds up 
into 34. The corresponding figure for the inscribed polygon 

js __?!„, which, he says, is > 3fT- This example shows 

2017J 
how little the Greeks were embarrassed in arithmetical 
calculations by their alphabetical system of numerals. 

On Conoids and Spheroids. 

The preface addressed to Dositheus shows, as we may 
also infer from internal evidence, that the whole of this 
book also was original. Archimedes first explains what 
his conoids and spheroids are, and then, after each 



40 ARCHIMEDES 

description, states the main results which it is the aim of 
the treatise to prove. The conoids are two. The first 
is the right-angled conoid, 3: TLdiVn^ adapted from the old 
name (** section of a right-angled cone") for a parabola ; 
this conoid is therefore a paraboloid of revolution. The 
second is the obtuse-angled conoid, which is a hyperbqloid 
of revolution described by the revolution bra hyperbola 
(a " section of an obtuse-angled cone ") about its trans- 
verse axis. The spheroids are two, being the solids of 
revolution described by the revolution of an ellipse (a 
"section of an acute-angled cone") about (i) its major 
axis and (2) its minor axis; the first is called the "ob- 
long " (or oblate) spheroid, the second the " flat " (or 
prolate) spheroid. As the volumes of oblique segments 
of conoids and spheroids are afterwards found in terms 
of the volume of the conical figure with the base of the 
segment as base and the vertex of the segment as vertex, 
and as the said base is thus an elliptic section of an 
oblique circular cone, Archimedes calls the conical figure 
with an elliptic base a " segment of a cone " as distinct 
from a " cone ". 

As usual, a series of preliminary propositions is re- 
quired. Archimedes first sums, in geometrical form, 
certain series, including the arithmetical progression, a, 
2a, ^a, . . . na, and the series formed by the squares of 
these terms (in other words the series i^, 2^, 3^ . . . n^) ; 
these summations are required for the final addition of 
an indefinite number of elements of each figure, which 
amounts to an inte^ral^ion. Next come two properties 
of conies (Prop. 3), then the determination by the method 
of exhaustion of the area of an ellipse (Prop. 4). Three 
propositions follow, the first two of which (Props. 7, 8) 
show that the conical figure above referred to is really 
a segment of an oblique circular cone ; this is done by 
actually finding the circular sections. Prop. 9 gives a 
similar proof that each elliptic section of a conoid or 
spheroid is a section of a certain oblique circular cylinder 



GEOMETRY IN ARCHIMEDES 41 

(with axis parallel to the axis of the segment of the 
conoid or spheroid cut off by the said elliptic section). 
Props, 1 1- 1 8 show the nature of the various sections 
which cut off segments of each condid and spheroid and 
which are circles or ellipses according as the section is 
perpendicular or obliquely inclined to the axis of the 
solid ; they include also certain properties of tangent 
planes, etc. 

The real business of the treatise begins with Props. 
19, 20; here it is shown how, by drawing many plane 
sections equidistant from one another and all parallel 
to the base of the segment of the solid, and describing 
cylinders (in general oblique) through each plane section 
with generators parallel to the axis of the segment and 
terminated by the contiguous sections on either side, we 
can make figures circumscribed and inscribed to the seg- 
ment, made up of segments of cylinders with parallel 
faces and presenting the appearance of the steps of a 
staircase. Adding the elements of the inscribed and 
circumscribed fig ares respectively and using the method 
of exhaustion, Archimedes finds the volumes of the re- 
spective segments of the solids in the approved manner 
(Props. 21, 22 for the paraboloid, Props. 25, 26 for the 
hyperboloid, and Props. 27-30 for the spheroids). The 
results are stated in this form: (i) Any segment of a 
paraboloid of revolution is half as large again as the cone 
or segment of a cone which has the same base and axis ; 
(2) Any segment of a hyperboloid of revolution or of a 
spheroid is to the cone or segment of a cone with the 
same base and axis in the ratio of AD + 3CA to AD + 2CA 
in the case of the hyperboloid, and of 3CA - AD to 
2CA- AD in the case of the spheroid, where C is the 
centre, A the vertex of the segment, and AD the axis 
of the segment (supposed in the case of the spheroid to 
be not greater than half the spheroid). 



42 ARCHIMEDES 

On Spirals. 

The preface addressed to Dositheus is of some length 
and contains, first, a tribute to the memory of Conon, 
and next a summary of the theorems about the sphere 
and the conoids and spheroids included in the above two 
treatises. Archimedes then passes to the spiral which, 
he says, presents another sort of problem, having nothing 
in common with the foregoing. After a definition of the 
spiral he enunciates the main propositions about it which 
are to be proved in the treatise. The spiral (now known 
as the Spiral of Archimedes) is defined as the locus of a 
point starting from a given point (called the "origin") 
on a given straight line and moving along the straight 
line at uniform speed, while the line itself revolves at 
uniform speed about the origin as a fixed point. Props. 
I -I I are preliminary, the last two amounting to the sum- 
mation of certain series required for the final addition 
of an indefinite number of element-areas, which again 
amounts to integration, in order to find the area of the 
figure cut off between any portion of the curve and the 
two radii vectores drawn to its extremities. Props. 13-20 
are interesting and difficult propositions establishing the 
properties of tangents to the spiral. Props. 21-23 show 
how to inscribe and circumscribe to any portion of the 
spiral figures consisting of a multitude of elements which 
are narrow sectors of circles with the origin as centre ; 
the area of the spiral is intermediate between the areas 
of the inscribed and circumscribed figures, and by the 
usual method of exhaustion Archimedes finds the areas 
required. Prop. 24 gives the area of the first complete 
turn of the spiral (= ^7r(27r^)^, where the spiral is r = aO), 
and of any portion of it up to OP where P is any point 
on the first turn. Props. 25, 26 deal similarly with the 
second turn of the spiral and with the area subtended by 
any arc (not being greater than a complete turn) on any 
turn. Prop. 27 proves the interesting property that, if 



GEOMETRY IN ARCHIMEDES 43 

Rj be the area of the first turn of the spiral bounded by 
the initial line, Rg the area of the ring added by the 
second complete turn, R3 the area of the ring added by the 
third turn, and so on, then R3 = 2R2, R^ = 3R2, R5 = 4R2, 
and so on to R„ = (« - i)R2, while R2 = 6R1. 

Quadrature of the Parabolar- 

The title of this work seems originally to have been 
On the Section of a Right-angled Cone and to have been 
changed after the time of Apollonius, who was the first 
to call a parabola by that name. The preface addressed 
to Dositheus was evidently the first communication from 
Archimedes to him after the death of Conon. It begins 
with a feeling allusion to his lost friend, to whom the 
treatise was originally to have been sent. It is in this 
preface that Archimedes alludes to the lemma used by 
earlier geometers as the basis of the method of exhaus- 
tion (the Postulate of Archimedes, or the theorem of 
Euclid X., i). He mentions as having been proved by 
m.eans of it (l) the theorems that the areas of circles are 
to one another in the duplicate ratio of their diameters, 
and that the volumes of spheres are in the triplicate 
ratio of their diameters, and (2) the propositions proved 
by Eudoxus about the volumes of a cone and a pyramid. 
No one, he says, so far as he is aware, has yet tried to 
square the segment bounded by a straight line and a 
section of a right-angled cone (a parabola) ; but he has 
succeeded in proving, by means of the same lemma, that 
the parabolic segment is equal to four-thirds of the 
triangle on the same base and of equal height, and he 
sends the proofs, first as "investigated" by means of 
mechanics and secondly as " demonstrated " by geometry. 
The phraseology shows that here, as in the Method, ' 
Archimedes regarded the mechanical investigation as 
furnishing evidence rather than proof of the truth of the 
proposition, pure geometry alone furnishing the absolute 
proof required. 



44 ARCHIMEDES 

The mechanical proof with the necessary preliminary 
propositions about the parabola (some of which are 
merely quoted, while two, evidently original, are proved, 
Props. 4, 5) extends down to Prop. 17; the geometrical 
proof with other auxiliary propositions completes the 
book (Props. 18-24). The mechanical proof recalls that 
of the Method in some respects, but is more elaborate in 
that the elements of the area of the parabola to be 
measured are not straight lines but narrow strips. -The 
figures inscribed and circumscribed to the segment are 
made up of such narrow strips and have a saw-like edge ; 
all the elements are trapezia except two, which are 
triangles, one in each figure. Each trapezium (or 
triangle) is weighed where it is against another area 
hung at a fixed point of an assumed lever ; thus the 
whole of the inscribed and circumscribed figures respec- 
tively are weighed against the sum of an indefinite number 
of areas all suspended from one point on the lever. The 
result is obtained by a real integration^ confirmed as 
usual by a proof by the method of exhaustion. 

The geometrical proof proceeds thus. Drawing in 
the segment the inscribed triangle with the same base 
and height as the segment, Archimedes next inscribes 
triangles in precisely the same way in each of the seg- 
ments left over, and proves that the sum of the two new 
triangles is \ of the original inscribed triangle. Again, 
drawing triangles inscribed in the same way in the four 
segments left over, he proves that their sum is \ of the 
sum of the preceding pair of triangles and therefore {^^ 
of the original inscribed triangle. Proceeding thus, we 
have a series of areas exhausting the parabolic segment. 
Their sum, if we denote the first inscribed triangle by A^ is 

/1{1 + i + {\Y + (i)' +....} 

Archimedes proves geometrically in Prop. 23 that the 

sum of this infinite series is ^ J, and then confirms by 

reductio ad absurdum the equality of the area of the 
parabolic segment to this area. 



CHAPTER V. 

THE SANDRECKONER. 

The Sandreckoner deserves a place by itself. It is not 
mathematically very important ; but it is an arithmetical 
curiosity which illustrates the versatility and genius of 
Archimedes, and it contains some precious details of 
the history of Greek astronomy which, coming from such 
a source and at first hand, possess unique authority. We 
wiU begin with the astronomical data. They are con- 
tained in the preface addressed to King Gelon of Syracuse, 
which begins as follows : — 

" There are some, King Gelon, who think that the 
number of the sand is infinite in multitude ; and I mean 
by the sand not only that which exists about Syracuse 
and the rest of Sicily but also that which is found in 
every region whether inhabited or uninhabited. Again, 
there are some who, without regarding it as infinite, yet 
think that no number has been named which is great 
enough to exceed its multitude. And it is clear that 
they who hold this view, if they imagined a mass made 
up of sand in other respects as large as the mass of the 
earth, including in it all the seas and the hollows of the 
earth filled up to a height equal to that of the highest of 
the mountains, would be many times further still from 
recognising that any number could be expressed which 
exceeded the multitude of the sand so taken. But I will 
try to show you, by means of geometrical proofs which 
you will be able to follow, that, of the numbers named 
by me and given in the work which I sent to Zeuxippus, 

(45) 4 



46 ARCHIMEDES 

some exceed not only the number of the mass of sand 
equal in size to the earth filled up in the way described, 
but also that of a mass equal in size to the universe. 

'* Now you are aware that ' universe * is the name given 
by most astronomers to the sphere the centre of which is 
the centre of the earth, while the radius is equal to the 
• straight line between the centre of the sun and the centre 
of the earth. This is the common account, as you have 
heard from astronomers. But Aristarchus of Samos 
brought out a book consisting of some hypotheses, in 
which the premises lead to the conclusion that the uni- 
verse is many times greater than that now so called. 
His hypotheses are that the fixed stars and the sun re- 
main unmoved, that the earth revolves about the sun in 
the circumference of a circle, the sun lying in the centre 
of the orbit, and that the sphere of the fixed stars, 
situated about the same centre as the sun, is so great 
that the circle in which he supposes the earth to revolve 
bears such a ratio to the distance of the fixed stars as 
the centre of the sphere bears to its surface." 

Here then is absolute and practically contemporary 
evidence that the Greeks, in the person of Aristarchus of 
Samos (about 310-230 B.C.), had anticipated Copernicus. 

By the last words quoted Aristarchus only meant to 
say that the size of the earth is negligible in comparison 
with the immensity of the universe. This, however, does 
not suit Archimedes's purpose, because he has to assume 
a definite size, however large, for the universe. Con- 
sequently he takes a liberty with Aristarchus. He says 
that the centre (a mathematical point) can have no ratio 
whatever to the surface of the sphere, and that we must 
therefore take Aristarchus to mean that the size of the 
earth is to that of the so-called ''universe" as the size 
of the so-called " universe " is to that of the real universe 
in the new sense. 

Next, he has to assume certain dimensions for the 
earth, the moon and the sun, and to estimate the angle 



THE SANDRECKONER 47 

subtended at the centre of the earth by the sun's diameter; 
and in each case he has to exaggerate the probable 
figures so as to be on the safe side. While therefore 
(he says) some have tried to prove that the perimeter 
of the earth is 300,000 stadia (Eratosthenes, his con- 
temporary, made it 252,000 stadia, say 24,662 miles, 
giving a diameter of about 7,850 miles), he will assume 
it to be ten times as great or 3,000,000 stadia. The 
diameter of the earth, he continues, is greater than that 
of the moon and that of the sun is greater than that of 
the earth. Of the diameter of the sun he observes that 
Eudoxus had declared it to be nine times that of the 
moon, and his own father, Phidias, had made it twelve 
times, while Aristarchus had tried to prove that the 
diameter of the sun is greater than eighteen times but 
less than twenty times the diameter of the moon (this 
was in the treatise of Aristarchus On the Sizes and Dis- 
tances of the Sun and Moon^ which is still extant, and 
is an admirable piece of geometry, proving rigorously, 
on the basis of certain assumptions, the result stated). 
Archimedes again intends to be on the safe side, so he 
takes the diameter of the sun to be thirty times that of 
the moon and not greater. Lastly, he says that Aris- 
tarchus discovered that the diameter of the sun appeared 
to be about y^th part of the zodiac circle, i.e. to sub- 
tend an angle of about half a degree ; and he describes 
a simple instrument by which he himself found that the 
angle subtended by the diameter of the sun at the time 
when it had just risen was less than xJjth part and 
greater than o-g^th part of a right angle. Taking this as 
the size of the angle subtended at the eye of the observer 
on the surface of the earth, he works out, by an interest- 
ing geometrical proposition, the size of the angle sub- 
tended at the centre of the earth, which he finds to 
be >^^'^^ part of a right angle. Consequently the 
diameter of the sun is greater than the side of a regular 
polygon of 812 sides inscribed in a great circle of the 



48 ARCHIMEDES 

so-called " universe," and a fortiori greater than the side 
of a regular chiliagon (polygon of looo sides) inscribed in 
that circle. 

On these assumptions, and seeing that the perimeter 
of a regular chiliagon (as of any other regular polygon 
of more than six sides) inscribed in a circle is more than 
3 times the length of the diameter of the circle, it easily 
follows that, while the diameter of the earth is less than 
1,000,000 stadia, the diameter of the so-called "uni- 
verse" is less than 10,000 times the diameter of the 
earth, and therefore less than 10,000,000,000 stadia. 

Lastly, Archimedes assumes that a quantity of sand 
not greater than a poppy-seed contains not more than 
10,000 grains, and that the diameter of a poppy-seed is 
not less than ^th of a dactylus (while a stadium is less 
than 10,000 dactyli). 

Archimedes is now ready to work out his calculation, 
but for the inadequacy of the alphabetic system of 
numerals to express such large numbers as are required. 
He, therefore, develops his remarkable terminology for 
expressing large numbers. 

The Greek has names for all numbers up to a myriad 
(10,000) ; there was, therefore, no difficulty in expressing 
with the ordinary numerals all numbers up to a myriad 
myriads (100,000,000). Let us, says Archimedes, call 
all these numbers numbers of the first order. Let the 
second order of numbers begin with 100,000,000, and end 
with ioo,ooo,oool Let 100,000,000-^ be the first number 
of the third order, and let this extend to 100,000,000^; 
and so on, to the myriad-niyriadth order, beginning with 
100,000,00099-999.999 and ending with 1 00,000,000 '°°'°°°'°~, 
which for brevity we will call P. Let all the numbers 
of all the orders up to P form the first period^ and let 
\ki^ first order oi ^'t second period begin with P and end 
with 100,000,000 P ; let the second order h^gin with this, 
the third order with 100,000,000^ P, and so on up to the 
joo^ooOyOOOth order of the second period, ending with 



THE SANDRECKONER 49 

1 00,000, ooo^*^'*^'"^ P or PI i:\\& first order of the third 
period begins with P^, and the orders proceed as before. 
Continuing the series o{ periods and orders of each period, 
we finally arrive at the 100, 000, 000th period ending with 
pioo,ooo.ooo^ The prodigious extent of this scheme is seen 
when it is considered that the last number of the first 
period would now be represented by i followed by 
800,000,000 ciphers, while the last number of the 
100,000,000th period would require 100,000,000 times 
as many ciphers, i.e. 80,000 million million ciphers. 

As a matter of fact, Archimedes does not need, in 
order to express the " number of the sand," to go beyond 
the eighth order of the first period. The orders of the 
;?r.y//m^<a^ begin respectively with i, 10^, 10^^, lO^^ ... 
( 1 08)99-999.999 ; and we can express all the numbers re- 
quired in powers of 10. 

Since the diameter of a poppy-seed is not less than 
4\^th of a dactylus, and spheres are to one another in the 
triplicate ratio of their diameters, a sphere of diameter 
I dactylus is not greater than 64,000 poppy- seeds, and, 
therefore, contains not more than 64,000 x 10,000 grains 
of sand, and a fortiori not more than 1,000,000,000, or 
I o9 grains of sand. Archimedes multiplies the diameter 
of the sphere continually by 100, and states the corre- 
sponding number of grains of sand A sphere of diame- 
ter 10,000 dactyli and a fortiori of one stadium contains 
less than 10^^ grains; and proceeding in this way to 
spheres of diameter 100 stadia, 10,000 stadia and so on, 
he arrives at the number of grains of sand in a sphere 
of diameter 10,000,000,000 stadia, which is the size of 
the so-called universe; the corresponding number of 
grains of sand is lo^^ The diameter of the real universe 
being 10,000 times that of the so-called universe, the 
final number of grains of sand in the real universe is 
found to be io^3^ which in Archimedes's terminology is a 
myriad-myriad units of the eighth order of numbers. 



CHAPTER VI. 

MECHANICS. 

It is said that Archytas was the first to treat mechanics 
in a systematic way by the aid of mathematical principles ; 
but no trace survives of any such work by him. In 
practical mechanics he is said to have constructed a 
mechanical dove which would fly, and also a rattle to 
amuse children and "keep them from breaking things 
about the house" (so says Aristotle, adding "for it is 
impossible for children to keep still"). 

In the Aristotelian Mechanica we find a remark on the 
marvel of a great weight being moved by a small force, 
and the problems discussed bring in the lever in various 
forms as a means of doing this. We are told also that 
practically all movements in mechanics reduce to the 
lever and the principle of the lever (that the weight and 
the force are in inverse proportion to the distances from 
the point of suspension or fulcrum of the points at which 
they act, it being assumed that they act in directions 
perpendicular to the lever). But the lever is merely 
"referred to the circle"; the force which acts at the 
greater distance from the fulcrum is said to move a weight 
more easily because it describes a greater circle. 

There is, therefore, no proof here. It was reserved for 
Archimedes to prove the property of the lever or balance 
mathematically, on the basis of certain postulates pre- 
cisely formulated and making no large demand on the 
faith of the learner. The treatise On Plane Equilibriums 

(50) 



^ MECHANICS 51 

in two books is, as the title implies, a work on statics 
only ; and, after the principle of the lever or balance has 
been established in Props. 6, 7 of Book I., the rest of 
the treatise is devoted to finding the centre of gravity of 
certain figures. There is no dynamics in the work and 
therefore no room for the parallelogram of velocities, 
which is given with a fairly adequate proof in the 
Aristotelian Mechanica. 

Archimedes's postulates include assumptions to the 
following effect: (i) Equal weights at equal distances 
are in equilibrium, and equal weights at unequal distances 
are not in equilibrium, but the system in that case 
*' inclines towards the weight which is at the greater 
distance," in other words, the action of the weight which 
is at the greater distance produces motion in the direc- 
tion in which it acts ; (2) and (3) If when weights are 
in equilibrium something is added to or subtracted from 
one of the weights, the system will "incline" towards 
the weight which is added to or the weight from which 
nothing is taken respectively ; (4) and (5) If equal and 
similar figures be applied to one another so as to coin- 
cide throughout, their centres of gravity also coincide ; if 
figures be unequal but similar, their centres of gravity 
are similarly situated with regard to the figures. 

The main proposition, that two magnitudes balance at 
distances reciprocally proportional to the magnitudes, is 
proved first for commensurable and then for incom- 
mensurable magnitudes. Preliminary propositions have 
dealt with equal magnitudes disposed at equal distances 
on a straight line and odd or even in number, and have 
shown where the centre of gravity of the whole system 
lies. Take first the case of commensurable magnitudes. 
If A, B be the weights acting at E, D on the straight 
line ED respectively, and ED be divided at C so that 
A : B = DC : CE, Archimedes has to prove that the 
system is in equilibrium about C. He produces ED to 
K, so that DK = EC, and DE to L so that EL = CD ; 



52 ARCHIMEDES 

LK is then a straight line bisected at C Again, let H 
be taken on LK such that LH = 2LE or 2CD, and 
it follows that the remainder HK = 2DK or 2EC. 
Since A, B are commensurable, so are EC, CD. Let 
;r be a common measuie of EC, CD. Take a weight 
w such that w is the same part of A that x is of LH. 
It follows that w is the same part of B that x is of HK. 
Archimedes now divides LH, HK into parts equal to x, 
and A B into parts equal to w. and places the w's at 
the middle points of the x'% respectively. All the w's 
are then in equilibrium about C. But all the ze/'s acting 
at the several points along LH are equivalent to A 
acting as a whole at the point E. Similarly the zf's 
acting at the several points on HK are equivalent to B 
acting at D. Therefore A, B placed at E, D respectively 
balance about C. 

Prop. 7 deduces by reductio ad absurdum the same 
result in the case where A, B are incommensurable. 
Prop. 8 shows how to find the centre of gravity of the 
remainder of a magnitude when the centre of gravity of 
the whole and of a part respectively are known. Props. 
9-15 find the centres of gravity of a parallelogram, a 
triangle and a parallel-trapezium respectively. 

Book II., in ten propositions, is entirely devoted to 
finding the centre of gravity of a parabolic segment, an 
elegant but difficult piece of geometrical work which is 
as usual confirmed by the method of exhaustion. 



CHAPTER VII. 

HYDROSTATICS. 

The science of hydrostatics is, even more than that of 
statics, the original creation of Archimedes. In hydro- 
statics he seems to have had no predecessors. Only one 
of the facts proved in his work On Floating Bodies^ in 
two books, is given with a sort of proof in Aristotle. 
This is the proposition that the surface of a fluid at rest 
is that of a sphere with its centre at the centre of the 
earth. 

Archimedes founds his whole theory on two postulates, 
one of which. comes at the beginning and the other after 
Prop. 7 of Book I. Postulate i is as follows : — 

** Let us assume that a fluid has the property that, if 
its parts lie evenly and are continuous, the part which is 
less compressed is expelled by that which is more com- 
pressed, and each of its parts is compressed by the fluid 
above it perpendicularly, unless the fluid is shut up in 
something and compressed by something else." 

Postulate 2 is: **Let us assume that any body which 
is borne upwards in water is carried along the perpen- 
dicular [to the surface] which passes through the centre 
of gravity of the body ". 

In Prop. 2 Archimedes proves that the surface of any 
fluid at rest is the surface of a sphere the centre of which 
is the centre of the earth. Props. 3-7 deal with the 
behaviour, when placed in fluids, of solids (i) just as 

(53) 



54 ARCHIMEDES 

heavy as the fluid, (2) lighter than the fluid, (3) heavier 
than the fluid. It is proved (Props. 5, 6) that, if the 
solid is lighter than the fluid, it will not be completely 
immersed but only so far that the weight of the solid 
will be equal to that of the fluid displaced, and, if it be 
forcibly immersed, the solid will be driven upwards by a 
force equal to the difference between the weight of the 
solid and that of the fluid displaced. If the solid is 
heavier than the fluid, it will, if placed in the fluid, 
descend to the*bottom and, if weighed in the fluid, the 
solid will be lighter than its true weight by the weight 
of the fluid displaced (Prop. 7). 

The last-mentioned theorem naturally connects itself 
with the story of the crown made for Hieron. It was 
suspected that this was not wholly of gold but contained 
an admixture of silver, and Hieron put to Archimedes 
the problem of determining the proportions in which the 
metals were mixed. It was the discovery of the solution 
of this problem when in the bath that made Archimedes 
run home naked, shouting €vp7]Ka, evprjKa. One account 
of the solution makes Archimedes use the proposition 
last quoted ; but on the whole it seems more likely that 
the actual discovery was made by a more elementary 
method described by Vitruvius. Observing, as he is said 
to have done, that, if he stepped into the bath when it 
was full, a volume of water was spilt equal to the volume 
of his body, he thought of applying the same idea to the 
case of the crown and measuring the volumes of water dis- 
placed respectively (i) by the crown itself, (2) by the same 
weight of pure gold, and (3) by the same weight of pure 
silver. This gives an easy means of solution. Suppose 
that the weight of the crown is W, and that it contains 
weights Wi and Wz of gold and silver respectively. Now 
experiment shows (i) that the crown itself displaces 
a certain volume of water, V say, (2) that a weight 
W of gold displaces a certain other volume of water. 



HYDROSTATICS 55 

Vi say, and (3) that a weight W of silver displaces a 

volume V^. 

From (2) it follows, by proportion, that a weight w^ of 

zv 
gold will displace -ttI . V^ of the fluid, and from (3) it 

follows that a weight w^ of silver displaces r^. V2of the 

fluid. 

Hence V=9.V,+ ^.V.; 

therefore W V = 'W^Vi + 'W2Y2> 

that is, (z^i + ^2)^ = ^1^1 + ^2V<„ 

so that wjw^ = (Vo - V)/(V - Y,)] 

which gives the required ratio of the weights of gold and 

silver contained in the crown. 

The last two propositions of Book I. investigate the 
case of a segment of a sphere floating in a fluid when the 
base of the segment is (i) entirely above and (2) entirely 
below the surface of the fluid ; and it is shown that the 
segment will in either case be in equilibrium in the posi- 
tion in which the axis is vertical, the equilibrium being 
in the first case stable. 

Book n. is a geometrical tour de force. Here, by the 
methods of pure geometry, Archimedes investigates the 
positions of rest and stability of a right segment of a 
paraboloid of revolution floating with its base upwards 
or downwards (but completely above or completely 
below the surface) for a number of cases differing (i) ac- 
cording to the relation between the length of the axis of 
the paraboloid and the principal parameter of the gene- 
rating parabola, and (2) according to the specific gravity 
of the solid in relation to the fluid ; where the position 
of rest and stability is such that the axis of the solid is 
not vertical, the angle at which it is inclined to the 
vertical is fully determined 

The idea of specific gravity appears all through, though 



56 ARCHIMEDES 

this actual term is not used. Archimedes speaks of the 
solid being lighter or heavier than the fluid or equally 
heavy with it, or, when a ratio has to be expressed, he 
speaks of a solid the weight of which (for an equal volume) 
has a certain ratio to that of the fluid. 



BIBLIOGRAPHY. 

The editio princeps of the works of Archimedes with the com- 
mentaries of Eutocius was brought out by Hervagius (Herwagen) 
at Basel in 1544. D. Rivault (Paris, 161 5) gave the enuncia- 
tions in Greek and the proofs in Latin somewhat retouched. 
The Arenarius {Sandreckoner) and the Dimensio circuit with 
Eutocius's commentary were edited with Latin translation and 
notes by Wallis in 1678 (Oxford). Torelli's monumental edi- 
tion (Oxford, 1792) of the Greek text of the complete works 
and of the commentaries of Eutocius, with a new Latin trans- 
lation, remained the standard text until recent years; it is 
now superseded by the definitive text with Latin translation of 
the complete works, Eutocius's commentaries, the fragments, 
scholia, etc., edited by Heiberg in three volumes (Teubner, 
Leipzig, first edition, 1 880-1; second edition,- including the 
newly discovered Method, etc., 191 0-15). 

Of translations the following may be mentioned. The Aldine 
edition of 1558, 4to, contains the Latin translation by Comman- 
dinus of the Measurement of a Circle, On Spirals, Quadrature 
of the Parabola, On Conoids and Spheroids, The Sandreckoner. 
Isaac Barrow's version was contained in Opera Archimedis, Apol- 
lonii Pergcei conicorum libri, Theodosii Sphcerica, methodo novo 
illustrata et demonstrata (London, 1675). The first French 
version of the works was by Peyrard in two volumes (second edi- 
tion, 1808). A valuable German translation, with notes, by E. 
Nizze, was published at Stralsund in 1824. There is a com- 
plete edition in modern notation by T. L. Heath (The Works 
of Archimedes, Cambridge, 1897, supplemented by The Method 
of Archimedes, Cambridge, 191 2). 



(57) 



{ 


CHRONOLOGY. 


(approximate in some cases.) 


B.C. 

624-547 


Thales 


572-497 


Pythagoras 


500-428 


Anaxagoras 


470-400 


f Hippocrates of Chios 


\ Hippias of Elis 


470-380 


Democritus 


460-385 


Theodorus of Cyrene 


430-360 


Archytas of Taras (Tarentum) 


427-347 


Plato 


415-369 


Theaetetus 


408-355 


Eudoxus of Cnidos 




Leon 
Menaechmus 




fl. about 350 


^ Dinostratus 
Theudius 




fl. 300 


Euclid 


310-230 


Aristarchus of Samos 


287-212 


Archimedes 


284-203 


Eratosthenes 


265-190 


Apollonius of Perga 



(58) 



f 



4^4 



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